(to be turned in on May 6 as part of WPR III Grade)

# Sequences of Functions and Series

**1.** Consider continuous functions from $[0,1]$ to $\mathbb{R}$ satisfying $0\leq f_1\leq f_2\leq f_3 \leq \cdots$ which converge pointwise to *f*. Must *f* be continuous? Give a proof or a counterexample.

**2.** Prove convergence and find the limit, or prove that the series does not converge:

**3.** What are the possible values of rearrangements of the following series?

# The Lipschitz Condition

A function is said to be *Lipschitz* with Lipschitz constant *C* if $|f(x)-f(y)|\leq C|x-y|$ for all *x,y* in the domain. (See Wikipedia article on Rudolf_Lipschitz.)

**4.** What does the Lipschitz condition say about the secant lines of *f*?

**5.** Prove that if *f* is Lipschitz on a particular domain, then *f* is uniformly continuous on that domain.

**6.** Prove that if the derivative of a function *f* exists and is bounded on some domain, that is $|f'(x)|\leq M$ for some *M*, then *f* is Lipschitz on that domain. How does the Lipschitz constant relate to the bound *M*?

**7.** Does the reverse hold true? That is, if *f* is Lipschitz, must the derivative (provided it exists) be bounded? Prove or give a counterexample.

**8.** Let $(f_n)$ be a pointwise convergent sequence of Lipschitz functions on $[0,1]$ with Lipschitz constant *C*. Prove that the sequence converges uniformly. Will the limit $\lim_{n\to\infty} f_n$ also be Lipschitz?

**9.** Let $x,y$ be fixed. Suppose *f* is Lipschitz with constant *C*. Define

What can you say about the limits of $f_n(x)$ and $f_n(y)$ for $n\to\infty$ if (a) $C<1$ or (b) $C\geq 1$?